20 common optimization problems (how to set up)
Get the file: https://www.patreon.com/blackpenredpen/posts/how-to-set-up-20-169957520/
Most students don't fail optimization on the calculus, they fail on the setup. Translating word problems into a single, solvable function is the ultimate bottleneck in Calculus 1. I worked through 20 optimization problems to cover every primary setup pattern you will encounter on an exam: 1. Distance minimization 2.Area and perimeter constraints 3.Cost and revenue optimization 4.Trigonometric setup models
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(Updated: September 2026 https://www.patreon.com/blackpenredpen)
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0:00 Intro
Q1, 0:53, Two numbers with difference 40 and minimum product
Q2, 4:15, Two positive numbers with product 100 and minimum sum
Q3, 6:18, Closest point on y=sqrt(x) to (4,0)
Q4, 9:42, Angle theta to minimize pulling force
Q5, 12:19, Production level to maximize total profit
Q6, 18:00, Rectangular garden with wire mesh and stone wall (min cost)
Q7, 22:54, Theater ticket price to maximize revenue
Q8, 28:42, Open-top box from a 10x16 cardboard sheet
Q9, 34:08, Postal box with square base (max volume, length+girth=108)
Q10, 38:08, Closed box with square base and fixed volume (min surface area)
Q11, 42:45, Open-top cylindrical container (min surface area)
Q12, 48:47, Closed cylindrical can with fixed surface area (max volume)
Q13, 51:57, Poster dimensions with top/bottom and side margins (min total area)
Q14, 56:11, Largest rectangle inscribed in an ellipse
Q15, 1:02:52, Rectangle under the curve y=e^(-x^2)
Q16, 1:06:13, Cylinder inscribed in a cone
Q17, 1:13:11, Norman window with fixed perimeter (max area)
Q18, 1:16:57, Rowing boat to shore (minimize travel time)
Q19, 1:25:00, Minimum length ladder reaching over a fence to a building
Q20, 1:31:43, Maximize the angle between wires from two poles
